2 Exploratory data analysis

Let’s load the Credit data and:

Use the View command to look at raw data.
Now select() only Balance, Limit, Income, Rating and Age variables. (We will be using Rating and Age in a forthcoming exercise)

# Write your code here
#

Let’s look at some summary statistics for the variables that we need for the problem at hand.

# Write your code here
#

Skim summary statistics
 n obs: 400 
 n variables: 3 

── Variable type:integer ────────────────────────────────────────────────────────────────────────
 variable missing complete   n    mean      sd  p0     p25    p50     p75
  Balance       0      400 400  520.01  459.76   0   68.75  459.5  863   
    Limit       0      400 400 4735.6  2308.2  855 3088    4622.5 5872.75
  p100     hist
  1999 ▇▃▃▃▂▁▁▁
 13913 ▅▇▇▃▂▁▁▁

── Variable type:numeric ────────────────────────────────────────────────────────────────────────
 variable missing complete   n  mean    sd    p0   p25   p50   p75   p100
   Income       0      400 400 45.22 35.24 10.35 21.01 33.12 57.47 186.63
     hist
 ▇▃▂▁▁▁▁▁

Let’s also look at _______________ as visual aids.

We observe for example:

The mean and median credit card balance are $520.01 and $459.50 respectively.
25% of card holders had debts of $68.75 or less.
The mean and median credit card limit are $4735.6 and $4622.50 respectively.
75% of these card holders had incomes of $57,470 or less.

Since our outcome variable Balance and the explanatory variables Limit and Income are numerical, we can and have to compute the correlation coefficient between pairs of these variables before we proceed to build a model.

Credit %>%
  select(Balance, Limit, Income) %>% 
  cor()

          Balance     Limit    Income
Balance 1.0000000 0.8616973 0.4636565
Limit   0.8616973 1.0000000 0.7920883
Income  0.4636565 0.7920883 1.0000000

Balance with Limit is 0.862. This indicates a strong positive linear relationship, which makes sense as only individuals with large credit limits can accrue large credit card balances.
Balance with Income is 0.464. This is suggestive of another positive linear relationship, although not as strong as the relationship between Balance and Limit.
As an added bonus, we can read off the correlation coefficient of the two explanatory variables, Limit and Income of 0.792. In this case, we say there is a high degree of collinearity between these two explanatory variables.

Note: Collinearity (or multicollinearity) is a phenomenon in which one explanatory variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy. So in this case, if we knew someone’s credit card Limit and since Limit and Income are highly correlated, we could make a fairly accurate guess as to that person’s Income. Or put loosely, these two variables provided redundant information. For now let’s ignore any issues related to collinearity and press on.

Let’s visualize the relationship of the outcome variable with each of the two explanatory variables in two separate plots:

To get a sense of the joint relationship of all three variables simultaneously through a visualization, let’s display the data in a 3-dimensional (3D) scatterplot, where

The numerical outcome variable $y$ Balance is on the z-axis (vertical axis)
The two numerical explanatory variables form the “floor” axes. In this case
- The first numerical explanatory variable $x_1$, Income is on of the floor axes.
- The second numerical explanatory variable $x_2$, Limit is on the other floor axis.

# draw 3D scatterplot
p <- plot_ly(data = Credit, z = ~Balance, x = ~Income, y = ~Limit, opacity = 0.6, color = Credit$Balance) %>%
  add_markers() 
p

Exercise

Conduct a new exploratory data analysis with the same outcome variable $y$ being Balance but with Rating and Age as the new explanatory variables $x_1$ and $x_2$. Remember, this involves three things:

Looking at the raw values
Computing summary statistics of the variables of interest.
Creating informative visualizations

What can you say about the relationship between a credit card holder’s balance and their credit rating and age?

2.1 Multiple regression

We now use a + to consider multiple explanatory variables. Here is the syntax:

model_name <- lm(y ~ x1 + x2 + ... +xn, data = data_frame_name)

Balance_model <- lm(Balance ~ Limit + Income, data = Credit)
Balance_model


Call:
lm(formula = Balance ~ Limit + Income, data = Credit)

Coefficients:
(Intercept)        Limit       Income  
  -385.1793       0.2643      -7.6633

# Or use one of the followings to see more info...

get_regression_table(Balance_model)

# A tibble: 3 x 7
  term      estimate std_error statistic p_value lower_ci upper_ci
  <chr>        <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept -385.       19.5       -19.8       0 -423.    -347.   
2 Limit        0.264     0.006      45.0       0    0.253    0.276
3 Income      -7.66      0.385     -19.9       0   -8.42    -6.91

#summary(Balance_model)

Write your model here:

How do we interpret these three values that define the regression plane?

Intercept: -$385.18 (rounded to two decimal points to represent cents). The intercept in our case represents the credit card balance for an individual who has both a credit Limit of $0 and Income of $0.
- In our data however, the intercept has limited (or no) practical interpretation as ….
Limit: $0.26. Now that we have multiple variables to consider, we have to add a caveat to our interpretation: Holding all the other variables fixed (Income, in this case), for every increase of one unit in credit Limit (dollars), there is an associated increase of on average $0.26 in credit card balance.
Income: -$7.66. Similarly, Holding all the other variables fixed (Limit, in this case),, for every increase of one unit in Income (in other words, $1000 in income), there is an associated decrease of on average $7.66 in credit card balance.

WAIT! Did something go wrong? Interpretation of the Income coefficient is alarming.

Recall in individual scatterplots that when considered, both Limit and Income had positive relationships with the outcome variable Balance. As card holders’ credit limits increased their credit card balances tended to increase as well, and a similar relationship held for incomes and balances. In the above multiple regression, however, the slope for Income is now -7.66, suggesting a negative relationship between income and credit card balance. What explains these contradictory results?

This is known as **Simpson’s Paradox**, a phenomenon in which a trend appears in several different groups of data but disappears or reverses when these groups are combined.

2.2 Observed/fitted values and residuals

As we did previously, let’s look at the fitted values and residuals.

regression_points <- get_regression_points(Balance_model)
regression_points

# A tibble: 400 x 6
      ID Balance Limit Income Balance_hat residual
   <int>   <dbl> <dbl>  <dbl>       <dbl>    <dbl>
 1     1     333  3606   14.9        454.   -121. 
 2     2     903  6645  106.         559.    344. 
 3     3     580  7075  105.         683.   -103. 
 4     4     964  9504  149.         986.    -21.7
 5     5     331  4897   55.9        481.   -150. 
 6     6    1151  8047   80.2       1127.     23.6
 7     7     203  3388   21.0        349.   -146. 
 8     8     872  7114   71.4        948.    -76.0
 9     9     279  3300   15.1        371.    -92.2
10    10    1350  6819   71.1        873.    477. 
# … with 390 more rows

2.2.1 Diagnostics (Residual plot)

Exercise

Fit a new linear regression using where Rating and Age are the new numerical explanatory variables $x_1$ and $x_2$.

Get information about the “best-fitting” line from the regression table by applying the get_regression_table() function. How do the regression results match up with the results from your exploratory data analysis in the previous exercise?

2.3 One numerical & one categorical explanatory variable

Let’s revisit the instructor evaluation data introduced in Ch 6.

Consider a modeling scenario with:

A numerical outcome variable $y$. As before, instructor evaluation score.
Two explanatory variables:
- A numerical explanatory variable $x_1$: in this case, their age.
- A categorical explanatory variable $x_2$: in this case, their binary gender.

2.3.1 Exploratory data analysis

Let’s reload the evals data and select() only the needed subset of variables. Note that these are different than the variables chosen in Chapter 6. Let’s given this the name evals_ch7.

evals_ch7 <- evals %>%
  select(score, age, gender)

Let’s look at the raw data values both by using View() and the glimpse() functions.
Let’s look at some summary statistics using the skim() function from the skimr package:

evals_ch7 %>% 
  skim()

Skim summary statistics
 n obs: 463 
 n variables: 3 

── Variable type:factor ─────────────────────────────────────────────────────────────────────────
 variable missing complete   n n_unique                top_counts ordered
   gender       0      463 463        2 mal: 268, fem: 195, NA: 0   FALSE

── Variable type:integer ────────────────────────────────────────────────────────────────────────
 variable missing complete   n  mean  sd p0 p25 p50 p75 p100     hist
      age       0      463 463 48.37 9.8 29  42  48  57   73 ▅▅▅▇▅▇▂▁

── Variable type:numeric ────────────────────────────────────────────────────────────────────────
 variable missing complete   n mean   sd  p0 p25 p50 p75 p100     hist
    score       0      463 463 4.17 0.54 2.3 3.8 4.3 4.6    5 ▁▁▂▃▅▇▇▆

Describe the output: Homework

let’s compute the correlation between two numerical variables we have score and age. Recall that correlation coefficients only exist between numerical variables.

evals_ch7 %>% 
  get_correlation(formula = score ~ age)

# A tibble: 1 x 1
  correlation
        <dbl>
1      -0.107

We observe that they are _______ and __________ correlated.

Now, let’s try to visualize the correlation.

Create a scatterplot of score over age. Use the binary gender variable to color the point with two colors. Add a regression line (or two?) in to your scatterplot.

ggplot(evals_ch7, aes(x = age, y = score, color = gender)) +
  geom_jitter() +
  labs(x = "Age", y = "Teaching Score", color = "Gender", title = "Graph using 'method = lm'") +
  geom_smooth(method = "lm", se = FALSE)

Say a couple of interesting things about the graph you’ve created.

2.3.2 Multiple regression: Parallel slopes model

Much like we started to consider multiple explanatory variables using the + sign in the previous section, let’s fit a regression model and get the regression table.

# A tibble: 3 x 7
  term       estimate std_error statistic p_value lower_ci upper_ci
  <chr>         <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept     4.48      0.125     35.8    0        4.24     4.73 
2 age          -0.009     0.003     -3.28   0.001   -0.014   -0.003
3 gendermale    0.191     0.052      3.63   0        0.087    0.294

The modeling equation for this scenario is: (Write the model you obtained)

Write the model for male:

Write the model for female:

Let’s create the scatterplot of score over age AGAIN. Use the binary gender variable to color the point with two colors. Add a regression lines in to your scatterplot BUT, this time use the model(s) we created.

Interpretaions of the coefficients:

$b_{male} = 0.1906$ is the average difference in teaching score that men get relative to the baseline of women.
Accordingly, the intercepts (which in this case make no sense since no instructor can have an age of 0) are :

for women: $b_0= 4.484$

for men: $b_0 +b_{male} = 4.484 + 0.191 = 4.675$
Both men and women have the same slope. In other words, in this model the associated effect of age is the same for men and women. So for every increase of one year in age, there is on average an associated decrease of $b_{age} = -0.009$ in teaching score.

Warning! ⚠

But wait!!!, why is the lines in the original scatterplot different than the lines in this one? What is going on?

What we have in the original plot is known as an interaction effect between age and gender.
Focusing on fitting a model for each of men and women, we see that the resulting regression lines are different.
Thus, gender appears to interact in different ways for men and women with the different values of age.

Once you nitice an interaction effect on the original scatterplot, you will have to create a model with interaction…(next section)

2.3.3 Multiple regression: Interaction model

We say a model has an interaction effect if the associated effect of one variable depends on the value of another variable. These types of models usually prove to be tricky to view on first glance because of their complexity. In this case, the effect of age will depend on the value of gender. (as was suggested by the different slopes for men and women in our visual exploratory data analysis)

Let’s fit a regression with an interaction term. We add an interaction term using the * sign. Let’s fit this regression and save it in score_model_interaction, then we get the regression table using the get_regression_table() function as before.

score_model_interaction <- lm(score ~ age + gender + age * gender, data = evals_ch7)
get_regression_table(score_model_interaction)

# A tibble: 4 x 7
  term           estimate std_error statistic p_value lower_ci upper_ci
  <chr>             <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept         4.88      0.205     23.8    0        4.48     5.29 
2 age              -0.018     0.004     -3.92   0       -0.026   -0.009
3 gendermale       -0.446     0.265     -1.68   0.094   -0.968    0.076
4 age:gendermale    0.014     0.006      2.45   0.015    0.003    0.024

The modeling equation for this scenario is (Writing the equation):

\[\hat{y} = b_0 + b_1 \cdot x_1 + b_2 \cdot x_2 + b_3 \cdot x_1 \cdot x_2\]

\[\hat{score} = 4.883 - 0.018 \cdot age - 0.446 \cdot 1_{Male}(x) + 0.014 \cdot age \cdot 1_{Male}(x) \]

Write the model for male:

\[\hat{score} = 4.883 - 0.018 \cdot age - 0.446 \cdot 1 + 0.014 \cdot age \cdot 1 \] \[\hat{score} = 4.437 - 0.004 \cdot age \]

Write the model for female:

\[\hat{score} = 4.883 - 0.018 \cdot age - 0.446 \cdot 0 + 0.014 \cdot age \cdot 0 \] \[\hat{score} = 4.883 - 0.018 \cdot age \]

Lecture Notes for Ch 7 of MD: Multiple Regression

Dr. Hasthika Rupasinghe

Mar 25, 2019 at 10:14:41 AM

1 Introduction